Difference between revisions of "2005 PMWC Problems/Problem T7"
(New page: ==Problem== Skipper’s doghouse has a regular hexagonal base that measures one metre on each side. Skipper is tethered to a 2-metre rope which is fixed to a vertex. What is the area of th...) |
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==Solution== | ==Solution== | ||
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| − | The big sector has a radius of 2 and is 2/3 of the total circle that it used to be a part of, and thus has an area of <math>4\pi | + | The big sector has a radius of 2 and is 2/3 of the total circle that it used to be a part of, and thus has an area of <math>4\pi\cdot\frac{2}{3}=\frac{8}{3}\pi</math>. The two little sectors each have radius 1 and are 1/6 the total area of their circle, so the area of each of those is <math>\frac{\pi}{6}</math>. Adding, we get <math>\frac{8}{3}\pi+\frac{\pi}{6}+\frac{\pi}{6}=3\pi</math>. Then approximate. |
==See also== | ==See also== | ||
| − | {{PMWC box|year=2005|num- | + | {{PMWC box|year=2005|num-b=T6|num-a=T8}} |
[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] | ||
Latest revision as of 22:28, 21 October 2022
Problem
Skipper’s doghouse has a regular hexagonal base that measures one metre on each side. Skipper is tethered to a 2-metre rope which is fixed to a vertex. What is the area of the region outside the doghouse that Skipper can reach? Calculate an approximate answer by using 
or 
.
Solution
The big sector has a radius of 2 and is 2/3 of the total circle that it used to be a part of, and thus has an area of 
. The two little sectors each have radius 1 and are 1/6 the total area of their circle, so the area of each of those is 
. Adding, we get 
. Then approximate.
See also
| 2005 PMWC (Problems) | ||
| Preceded by Problem T6 |
Followed by Problem T8 | |
| I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
